Specific Gaussian mixtures are considered to solve simultaneously
variable selection and clustering problems. A non asymptotic
penalized criterion is proposed to choose the number of mixture
components and the relevant variable subset. Because of the non
linearity of the associated Kullback-Leibler contrast on Gaussian
mixtures, a general model selection theorem for maximum likelihood
estimation proposed by [Massart Concentration inequalities and model selection
Springer, Berlin (2007).
Lectures from the 33rd Summer School on Probability Theory held in
Saint-Flour, July 6–23 (2003)]
is used to obtain
the penalty function form. This theorem requires to control the
bracketing entropy of Gaussian mixture families. The ordered and
non-ordered variable selection cases are both addressed in this
paper.

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